Theory of Cryptography - Canetti, Ran (Springer, 2017)
A probabilistically checkable proof (PCP) system enables proofs to be verified in time polylogarithmic in the length of a classi-cal proof. Computationally sound (CS) proofs improve upon PCPs by additionally shortening the length of the transmitted proof to be poly-logarithmic in the length of the classical proof. In this paper we explore the ultimate limits of non-interactive proof systems with respect to time and space efficiency. We present a proof system where the prover uses space polynomial in the space of a classical prover and time essentially linear in the time of a classical prover, while the verifier uses time and space that are essentially constant. Further, this proof system iscomposable: there is an algorithm for merging two proofs of lengthkinto a proof of the conjunction of the original two theorems in time polynomial ink, yielding a proof of length exactlyk. We deduce the existence of our proposed proof system by way of a natural new assumption about proofs of knowledge. In fact, a main contribution of our result is showing that knowledge can be “traded” for time and space efficiency in noninteractive proof systems. We motivate this result with an explicit construction of noninteractive CS proofs of knowledge in the random oracle model.